1. Technical Field of the Invention
The present invention relates to the decoding of digital data transmitted over a communications channel and, in particular, to a low complexity maximum likelihood decoder for second order Reed-Muller codes.
2. Description of Related Art
There exist many applications where large volumes of digital data must be transmitted and received in a substantially error free manner. In telecommunications systems, in particular, it is imperative that the reception of digital data be accomplished as reliably as is possible, however, because the communications channels (including radio frequency, fiber optic, coaxial cable, and twisted copper wire) utilized for data transmission are plagued by error introducing factors. For example, such errors may be attributable to transient conditions in the channel (like interference, noise or multi-path fading). The influence of such factors results in instances where the digital data is not transmitted properly or cannot be reliably received.
Considerable attention has been directed toward overcoming this problem and reducing the number of errors incurred when transmitting data. One option involves increasing transmitter power. However, this is typically not practical due to limitations regarding transmitter electronics, regulations on peak power transmission, and the added expense involved in increasing power levels. A preferable alternative option for combating noise on the communications channel is to introduce redundancy in the transmitted message which is used at the receiver to correct introduced errors. Such redundancy is typically implemented through the use of error control coding (channel codes). A preferable alternative option for combating fading on the communications channel is to use an interleaver to reorder the data prior to transmission over the channel. As a result, many communications systems now utilize a combination of error control coding/decoding and interleaving/de-interleaving processes to protect against the effects of interference, noise or multi-path fading on the communications channel.
Because of implementation complexity concerns, the error control decoder typically used comprises a soft decision decoder (and, in particular, an errors and erasures decoder). Such decoders exploit reliability values output from a demodulator in estimating the transmitted codeword. In the absence of fading, and in the presence of Gaussian noise, the optimal soft decision decoder is the maximum likelihood decoder. It is also typically the best decoder in the presence of fading (assuming a good estimate of the fading is available). For a general block code, however, maximum likelihood decoding can be hopelessly complex to implement. Accordingly, a need exists for a less complex maximum likelihood decoding scheme for implementation in connection with soft decision decoding of block codes.
For the special case of the (24,12) Golay code and the (23,12) extended Golay code, a maximum likelihood decoder having a very low complexity has been devised by Conway and Sloane (see, IEEE Trans. Infor. Theory, vol. 32, pp. 41-50, 1986). The premise behind the Conway-Sloane decoding method is that for a given Golay code, an attempt is made to find a subcode of that given Golay code that is easy to decode. The given Golay code may then be decoded by cycling, to achieve a lower overall complexity, over the subcode and its cosets. For the (24,12) Golay code, for example, it is noted that there is a subcode thereof which is equivalent to a parity check code. Such a parity check code presents a trivial decoding challenge. A more complete explanation of the operation of the Conway-Sloane decoding method may be obtained by referring to the previously mentioned IEEE article, or to U.S. application for patent Ser. No. 08/768,530, filed Dec. 18, 1996, by Ali S. Khayrallah, et al., the disclosures of which are hereby incorporated by reference.
The disclosed Conway-Sloane decoding method is limited on its face to application in connection with the Golay and extended Golay codes. Furthermore, an extension of the Conway-Sloane decoding method has been proposed by the above-referenced U.S. Application for Patent in decoding various shortened (19,7), (18,7) and (18,6) Golay codes. In spite of these advances relating to more efficient decoding of Golay codes, a need still exists for less complex maximum likelihood decoding schemes specifically addressing other types of codes. In particular, there is a need for such a scheme in connection with the decoding of second order Reed-Muller codes.